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Theorem curry1val 7792
Description: The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
curry1.1 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
Assertion
Ref Expression
curry1val ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → (𝐺𝐷) = (𝐶𝐹𝐷))

Proof of Theorem curry1val
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 curry1.1 . . . 4 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
21curry1 7791 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
32fveq1d 6665 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → (𝐺𝐷) = ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷))
4 eqid 2819 . . . . . . 7 (𝑥𝐵 ↦ (𝐶𝐹𝑥)) = (𝑥𝐵 ↦ (𝐶𝐹𝑥))
54fvmptndm 6791 . . . . . 6 𝐷𝐵 → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
65adantl 484 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ ¬ 𝐷𝐵) → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
7 fndm 6448 . . . . . . 7 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
87adantr 483 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → dom 𝐹 = (𝐴 × 𝐵))
9 simpr 487 . . . . . . 7 ((𝐶𝐴𝐷𝐵) → 𝐷𝐵)
109con3i 157 . . . . . 6 𝐷𝐵 → ¬ (𝐶𝐴𝐷𝐵))
11 ndmovg 7323 . . . . . 6 ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐶𝐴𝐷𝐵)) → (𝐶𝐹𝐷) = ∅)
128, 10, 11syl2an 597 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ ¬ 𝐷𝐵) → (𝐶𝐹𝐷) = ∅)
136, 12eqtr4d 2857 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ ¬ 𝐷𝐵) → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
1413ex 415 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → (¬ 𝐷𝐵 → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷)))
15 oveq2 7156 . . . 4 (𝑥 = 𝐷 → (𝐶𝐹𝑥) = (𝐶𝐹𝐷))
16 ovex 7181 . . . 4 (𝐶𝐹𝐷) ∈ V
1715, 4, 16fvmpt 6761 . . 3 (𝐷𝐵 → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
1814, 17pm2.61d2 183 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
193, 18eqtrd 2854 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → (𝐺𝐷) = (𝐶𝐹𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1530  wcel 2107  Vcvv 3493  c0 4289  {csn 4559  cmpt 5137   × cxp 5546  ccnv 5547  dom cdm 5548  cres 5550  ccom 5552   Fn wfn 6343  cfv 6348  (class class class)co 7148  2nd c2nd 7680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-1st 7681  df-2nd 7682
This theorem is referenced by:  nvinvfval  28409  hhssabloilem  29030
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